A liar who always lies says "All my hats are green."

Original Article ↗ Hacker News Discussion ↗

A classic logic puzzle about a “liar who always lies” claiming “All my hats are green” prompts debate over what, if anything, can be deduced about the liar’s hats. Many apply formal logic and vacuous truth: if “all my hats are green” must be false, then there must exist at least one hat and at least one of them is not green, making “the liar has at least one hat” the only necessarily true option. Others argue that ordinary English strongly implies the speaker has hats and uses “all” differently from the mathematical ∀ quantifier, highlighting the gap between formal logic, programming intuitions, and everyday language.

Formal-logic reading and the “correct” answer

  • Many participants translate “All my hats are green” into a universal quantification over the set of the liar’s hats.
  • Under classical logic, a universal statement over an empty set is vacuously true.
  • Since the liar “only says false statements”, the utterance must be false, so the set of hats cannot be empty.
  • Negating “All my hats are green” yields “There exists at least one of my hats that is not green.”
  • From this, people conclude:
    • The liar has at least one hat.
    • At least one of those hats is not green.
    • Among the multiple-choice options, only “The liar has at least one hat” is necessarily true.

Colloquial / linguistic objections

  • Others argue that in everyday English, “all my X” strongly implies “I have at least one X”, so with zero hats the statement is already a lie.
  • On that reading, the lie could be:
    • about existence (having no hats), or
    • about color (having non‑green hats), or both.
  • Therefore none of the answer choices are forced; multiple scenarios are consistent.
  • This leads into discussion of pragmatics and implicature: natural language users expect relevance and informativeness, not vacuous truths.

Programming analogies & vacuous truth

  • Several comments relate the puzzle to programming constructs:
    • Folding AND/OR over empty lists and the role of identity elements.
    • Functions like all/every returning true on empty collections.
  • Some see this as strong intuition for vacuous truth; others insist empty-set semantics should be explicitly modeled, especially when NULL/“no data” is meaningful.

Critiques of the puzzle and logic-puzzle culture

  • Some call the puzzle a “gotcha” or “midwit trap” that relies on an unspoken mapping from English to formal logic.
  • Others reply that this is precisely the point: to expose the gap between social intuition and formal reasoning.
  • There is broader debate about:
    • Whether such puzzles test reasoning or just familiarity with specific conventions.
    • How much weight to give formal logic when interpreting natural language.

Related topics and extensions

  • The thread branches into related paradoxes and puzzles (Monty Hall, ravens, liar paradox, two-envelope problems, “always lies” gods), used to illustrate similar tensions between intuitive and formal reasoning.